Projections in Krein spaces (Q1039744)

The problem when the sum of two regular subspaces of a Krein space \((\mathcal K,[\cdot,\cdot])\) is again regular is treated. A regular subspace in a Krein space is a closed subspace which is, equipped with the indefinite inner product \([\cdot,\cdot]\), again a Krein space, or, what is the same, the range of a \(J\)-selfadjoint projection. Here, a \(J\)-selfadjoint projection \(P\) is a bounded linear idempotent operator (\(P^2=P\)) which is selfadjoint with respect to the indefinite inner product \([\cdot,\cdot]\). It is shown that the sum of a uniformly positive closed subspace and a uniformly negative closed subspace is always regular. Moreover, it is shown that every \(J\)-selfadjoint projection \(P\) can be written as the sum of a \(J\)-positive projection \(P_1\) and a \(J\)-negative projection \(P_2\) such that \(P_1\) commutes with \(P_2\) and with \(P_2^*\). Furthermore, the problem when the closure of the range of the product of a \(J\)-bicontraction \(A\) and a \(J\)-biexpansion \(B\) becomes regular is discussed. If \(A\) commutes with \(B\) and \(B^+\), where \(B^+\) denotes the adjoint of \(B\) with respect to the indefinite inner product \([\cdot,\cdot]\), then the closure of the range of \(AB\) is regular.